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5.3 Scientific Notation
Use a scientific calculator to compute the following: 5,000,000 × 6,000,000 = ___________________
Very large and very small numbers often occur in the sciences. For such numbers, we use
scientific notation simply because it’s easier to work with AND most calculators cannot display
enough digits to give the answer in decimal form!
The form for scientific notation is:
a  10n where 1  a  10 and n is an integer.
Example 1: Determine whether or not each number is written in scientific notation.
Circle the numbers written in scientific notation.
1.3  102
2  103
0.4  102
53.2  106
21  103
5.99 1046
Observe the following:
1
2.14 × 10−1 = 2.14 × 10 = 0.214
2.14 × 101 = 2.14 × 10 = 21.4
1
2.14 × 102 = 2.14 × 100 = 214
1
2.14 × 103 = 2.14 × 1000 = 2,140
2.14 × 10−2 = 2.14 × 100 = 0.0214
2.14 × 10−3 = 2.14 × 1000 = 0.00214
Do you notice a pattern??
How to write a number in decimal notation (without exponents):
For numbers greater than 10, the exponent n is positive and equal to the number of places the
decimal point in the number a moves to the right.
scientific notation
decimal notation (without exponents)
3.45 × 104 = 3. 4 5 0 0 . = 34,500
For numbers less than 1, the exponent n is negative and equal to the number of places the
decimal point in the number a moves to the left.
scientific notation
decimal notation (without exponents)
3.45 × 10−4 = . 0 0 0 3. 45 = .000345
Example 2: Write in decimal notation:
1. 7.4  108
_____________________
4. 2  102
_____________________
2. 3.54  106 _____________________
5. 1.333  104 _____________________
3. 2.5  102 _____________________
6. 8  101
_____________________
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How to write a number in scientific notation:
First write 𝑎 (where 1 ≤ 𝑎 < 10) by placing the decimal after the first nonzero digit.
For numbers greater than 10, the exponent n on the base 10 should be positive and equal to the
number of places the decimal point in 𝑎 would need to move to the right to yield the number
without exponents.
decimal notation (without exponents)
scientific notation
34,500 = 3. 4 5 0 0 . = 3.45 × 104
For numbers less than 1, the exponent n on the base 10 should be negative and equal to the
number of places the decimal point in 𝑎 would need to move to the left to yield the number
without exponents.
decimal notation (without exponents)
scientific notation
0.000345 = . 0 0 0 3. 45 = 3.45 × 10−4
Example 3: Write each number in scientific notation:
1. 0.00035
_____________________
4. 280,000
_____________________
2. 358
_____________________
5. 0.125
_____________________
3. 0.0000056 _____________________
6. 43,000,000 _____________________
Example 4: Perform the indicated operations. Write each answer (a) in scientific notation
and (b) without exponents.
1.
2.
4 × 107 (3 × 10−3 )
3×10 9
6×10 5